tutorial.md

Tutorial: Verifying the Bakery Algorithm

This tutorial explains how to verify the safety of Lamport's Bakery Algorithm using the DMC-Logic framework. We will walk through encoding the system transitions, defining inductive invariants, and proving mutual exclusion.

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1. Representing Rewrite Theories

In DMC-Logic, the system state is represented by a configuration Conf. The transitions between states are defined using an inductive proposition Step.

The Bakery algorithm uses a global ticket counter (t) and a serving counter (s). Processes move through states: idle → wait → crit.

inductive Step : Conf → Conf → Prop where
  -- A process picks a ticket (t) and moves to wait state
  | wake : ∀ n m : Nat, ∀ ps : ProcSet,
      Step {t := n,      s := m, c := {$[idle]}   + ps}
           {t := succ n, s := m, c := {$[wait n]} + ps}

  -- A process enters the critical section if its ticket matches the serving counter (s)
  | crit : ∀ n m : Nat, ∀ ps : ProcSet,
      Step {t := n, s := m, c := {$[wait m]} + ps}
           {t := n, s := m, c := {$[crit m]} + ps}

  -- A process exits, incrementing the serving counter
  | exit : ∀ n m : Nat, ∀ ps : ProcSet,
      Step {t := n, s := m,      c := {$[crit m]} + ps}
           {t := n, s := succ m, c := {$[idle]} + ps}

infix:110 " ⇒ " => Step
infix:110 " ⇒* " => Relation.ReflTransGen Step

## 2. Representing Inductive Invariants
To prove safety, we define predicates that describe all "reachable" and "safe" states. We categorize the system state into three main phases:

### Initial State
The system starts with the ticket and serving counters equal, and all processes `idle`.

```lean
def init_pred (cf : Conf) : Prop :=
  cf.t = cf.s ∧ is cf.c

Waiting State

When processes are waiting, their tickets must be between the current serving counter and the next available ticket.

def wait_pred (cf : Conf) : Prop :=
  cf.t > cf.s ∧ ws cf.c ∧ 
  (∀ k ∈ tickets cf.c, k ≥ cf.s ∧ k < cf.t) ∧ 
  ((tickets cf.c).Nodup)

Critical State

If a process is in the critical section, it must hold the ticket exactly matching cf.s.

def crit_pred (cf : Conf) : Prop :=
  cf.t > cf.s ∧
  ∃ ps : ProcSet,
    cf.c = {$[crit cf.s]} + ps ∧
    ws ps ∧
    (∀ k ∈ tickets ps, k > cf.s ∧ k < cf.t) ∧ 
    ((tickets ps).Nodup)

The Global Invariant

The overall invariant Inv is the disjunction of these three possibilities.

def Inv (cf : Conf) : Prop :=
  init_pred cf ∨ wait_pred cf ∨ crit_pred cf

3. Verifying Safety by Inductive Invariants

Verification in bakery.lean follows the standard inductive approach. We must prove two main properties:

1. base step: The starting state satisfies Inv.
2. inductive step: Every valid Step preserves Inv.

Base Step

We prove that the initial configuration (where t=0, s=0, and all processes are idle) satisfies init_pred.

theorem inv_step : ∀ cf cf' : Conf, Inv cf → cf ⇒ cf' → Inv cf' := by
  intros cf cf' h_inv h_step
  cases h_step <;> cases h_inv
  -- The proof proceeds by analyzing how each transition 
  -- affects the ticket ranges and the Nodup property.

Inductive Step

For every transition in Step (wake, crit, exit), we show that if Inv cf holds, then Inv cf must also hold.

theorem inv_step : ∀ cf cf' : Conf, Inv cf → cf ⇒ cf' → Inv cf' := by
  intros cf cf' h_inv h_step
  cases h_step <;> cases h_inv
  -- The proof proceeds by analyzing how each transition 
  -- affects the ticket ranges and the Nodup property.

4. Final Safety Result: Mutual Exclusion

The ultimate goal is to prove Mutual Exclusion: no two processes can be in the crit state simultaneously. Once we have proven that Inv is inductive, we derive the final safety theorem:

theorem safety : ∀ cf : Conf, init_conf ⇒* cf → Inv cf := by
  apply Relation.ReflTransGen.connective_property
  . apply inv_init
  . apply inv_step

-- This ensures that it is impossible for two different processes 
-- to be in the critical section at the same time.
theorem mutual_exclusion : ∀ cf : Conf, Inv cf → 
  (∀ m n, {$[crit m]} + {$[crit n]} ⊆ cf.c → m = n) := by
  -- Follows from the structure of crit_pred

5. Summary

By using DMC-Logic, we transformed the Bakery Algorithm into a formal rewrite theory. We defined an inductive invariant that captures the relationship between the ticket counters and the process states, and used Lean to mechanically verify that this invariant holds across all possible executions.